Tan Inverse Calculator
Calculate inverse tangent (arctan) with detailed explanations and special value recognition
Calculate Inverse Tangent (arctan)
Any real number (domain: all reals)
Choose output angle unit
Inverse Tangent Results
Calculation Details
Formula: y = arctan(x) where x = 0
In radians: arctan(0) = 0.000000 rad
In degrees: arctan(0) = 0.000000°
Verification: tan(0.000000°) = 0.000000
Related Trigonometric Values
Function Properties
Domain: All real numbers
Range: (-π/2, π/2) or (-90°, 90°)
Function type: Odd function
Monotonicity: Strictly increasing
Example Calculations
Special Values
arctan(0) = 0° (exact value)
arctan(1) = 45° (exact value)
arctan(-1) = -45° (exact value)
arctan(√3) = 60° (exact value)
arctan(1/√3) = 30° (exact value)
Right Triangle Example: arctan(1)
Step 1: Consider a right triangle where opposite = adjacent
Step 2: This means tan(θ) = opposite/adjacent = 1
Step 3: This triangle is half of a square cut along diagonal
Step 4: The angle is half of 90°, which is 45°
Result: arctan(1) = 45°
Understanding the Range
Output range: (-90°, 90°) or (-π/2, π/2) radians
Positive inputs: Give angles in Quadrant I (0° to 90°)
Negative inputs: Give angles in Quadrant IV (-90° to 0°)
Zero input: Gives exactly 0°
Arctan Function Properties
Domain
All real numbers (-∞, ∞)
Range
(-π/2, π/2) or (-90°, 90°)
Symmetry
Odd function: arctan(-x) = -arctan(x)
Monotonicity
Strictly increasing everywhere
Special Values
Quick Tips
Arctan is the inverse of tangent function
Domain: all real numbers
Range: (-90°, 90°) or (-π/2, π/2)
Always gives principal value
Used to find angles from ratios
Understanding Inverse Tangent (arctan)
What is Arctan?
The inverse tangent function, denoted as arctan(x) or tan⁻¹(x), is the inverse of the tangent function. It answers the question: "What angle has a tangent value of x?"
y = arctan(x)
means tan(y) = x
Key Characteristics
- •Domain: All real numbers
- •Range: (-90°, 90°) or (-π/2, π/2)
- •Odd function: arctan(-x) = -arctan(x)
- •Continuous: No breaks or jumps
Applications
- •Trigonometry: Finding angles in right triangles
- •Physics: Calculating angles of trajectory and forces
- •Engineering: Slope calculations and angle measurements
- •Navigation: Bearing and heading calculations
Why the Restricted Range?
The tangent function repeats every 180° (π radians), so it's not one-to-one over all real numbers. To create an inverse function, we restrict the domain of tangent to (-90°, 90°) where it's strictly increasing.
Note: This means arctan always returns the "principal value" - the angle in the range (-90°, 90°).